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Theorem txindis 22158
Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindis ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Proof of Theorem txindis
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 4313 . . . . . . 7 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
2 indistop 21526 . . . . . . . . . . 11 {∅, 𝐴} ∈ Top
3 indistop 21526 . . . . . . . . . . 11 {∅, 𝐵} ∈ Top
4 eltx 22092 . . . . . . . . . . 11 (({∅, 𝐴} ∈ Top ∧ {∅, 𝐵} ∈ Top) → (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
52, 3, 4mp2an 688 . . . . . . . . . 10 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
6 rsp 3210 . . . . . . . . . 10 (∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → (𝑦𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
75, 6sylbi 218 . . . . . . . . 9 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
8 elssuni 4866 . . . . . . . . . . . . . 14 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ({∅, 𝐴} ×t {∅, 𝐵}))
9 indisuni 21527 . . . . . . . . . . . . . . 15 ( I ‘𝐴) = {∅, 𝐴}
10 indisuni 21527 . . . . . . . . . . . . . . 15 ( I ‘𝐵) = {∅, 𝐵}
112, 3, 9, 10txunii 22117 . . . . . . . . . . . . . 14 (( I ‘𝐴) × ( I ‘𝐵)) = ({∅, 𝐴} ×t {∅, 𝐵})
128, 11sseqtrrdi 4022 . . . . . . . . . . . . 13 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
1312ad2antrr 722 . . . . . . . . . . . 12 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
14 ne0i 4304 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑧 × 𝑤) → (𝑧 × 𝑤) ≠ ∅)
1514ad2antrl 724 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ≠ ∅)
16 xpnz 6014 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅) ↔ (𝑧 × 𝑤) ≠ ∅)
1715, 16sylibr 235 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅))
1817simpld 495 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ≠ ∅)
1918neneqd 3026 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑧 = ∅)
20 simpll 763 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, 𝐴})
21 indislem 21524 . . . . . . . . . . . . . . . . . . 19 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2220, 21syl6eleqr 2929 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, ( I ‘𝐴)})
23 elpri 4586 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {∅, ( I ‘𝐴)} → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
2422, 23syl 17 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
2524ord 860 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑧 = ∅ → 𝑧 = ( I ‘𝐴)))
2619, 25mpd 15 . . . . . . . . . . . . . . 15 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 = ( I ‘𝐴))
2717simprd 496 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ≠ ∅)
2827neneqd 3026 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑤 = ∅)
29 simplr 765 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, 𝐵})
30 indislem 21524 . . . . . . . . . . . . . . . . . . 19 {∅, ( I ‘𝐵)} = {∅, 𝐵}
3129, 30syl6eleqr 2929 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, ( I ‘𝐵)})
32 elpri 4586 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {∅, ( I ‘𝐵)} → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
3331, 32syl 17 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
3433ord 860 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑤 = ∅ → 𝑤 = ( I ‘𝐵)))
3528, 34mpd 15 . . . . . . . . . . . . . . 15 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 = ( I ‘𝐵))
3626, 35xpeq12d 5585 . . . . . . . . . . . . . 14 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) = (( I ‘𝐴) × ( I ‘𝐵)))
37 simprr 769 . . . . . . . . . . . . . 14 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ⊆ 𝑥)
3836, 37eqsstrrd 4010 . . . . . . . . . . . . 13 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
3938adantll 710 . . . . . . . . . . . 12 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
4013, 39eqssd 3988 . . . . . . . . . . 11 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵)))
4140ex 413 . . . . . . . . . 10 ((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4241rexlimdvva 3299 . . . . . . . . 9 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
437, 42syld 47 . . . . . . . 8 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦𝑥𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4443exlimdv 1927 . . . . . . 7 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑦 𝑦𝑥𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
451, 44syl5bi 243 . . . . . 6 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (¬ 𝑥 = ∅ → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4645orrd 859 . . . . 5 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
47 vex 3503 . . . . . 6 𝑥 ∈ V
4847elpr 4587 . . . . 5 (𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ↔ (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4946, 48sylibr 235 . . . 4 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))})
5049ssriv 3975 . . 3 ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ {∅, (( I ‘𝐴) × ( I ‘𝐵))}
519toptopon 21441 . . . . . . 7 ({∅, 𝐴} ∈ Top ↔ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)))
522, 51mpbi 231 . . . . . 6 {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
5310toptopon 21441 . . . . . . 7 ({∅, 𝐵} ∈ Top ↔ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵)))
543, 53mpbi 231 . . . . . 6 {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))
55 txtopon 22115 . . . . . 6 (({∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) ∧ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))) → ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))))
5652, 54, 55mp2an 688 . . . . 5 ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵)))
57 topgele 21454 . . . . 5 (({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))) → ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵))))
5856, 57ax-mp 5 . . . 4 ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵)))
5958simpli 484 . . 3 {∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵})
6050, 59eqssi 3987 . 2 ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (( I ‘𝐴) × ( I ‘𝐵))}
61 txindislem 22157 . . 3 (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
6261preq2i 4672 . 2 {∅, (( I ‘𝐴) × ( I ‘𝐵))} = {∅, ( I ‘(𝐴 × 𝐵))}
63 indislem 21524 . 2 {∅, ( I ‘(𝐴 × 𝐵))} = {∅, (𝐴 × 𝐵)}
6460, 62, 633eqtri 2853 1 ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843   = wceq 1530  wex 1773  wcel 2107  wne 3021  wral 3143  wrex 3144  wss 3940  c0 4295  𝒫 cpw 4542  {cpr 4566   cuni 4837   I cid 5458   × cxp 5552  cfv 6352  (class class class)co 7148  Topctop 21417  TopOnctopon 21434   ×t ctx 22084
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-fv 6360  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7680  df-2nd 7681  df-topgen 16707  df-top 21418  df-topon 21435  df-bases 21470  df-tx 22086
This theorem is referenced by: (None)
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